Search

Surrogate models are indispensable tools in uncertainty quantification and global sensitivity analysis. Polynomial chaos expansion (PCE) is one of the most widely used surrogate models, thanks to its faster convergence rate compared to Monte Carlo simulation. In some cases, especially for complex problems, analyzing the complexity of the random input–output relationship (e.g., nonlinearity and interactions between input variables) may reveal additional information and useful insight. To that end, this paper introduces the use of Shapley additive explanations (SHAP) to help the explanation of a PCE model. Originating from game theory and machine learning, SHAP computes the contribution of the input variables to the single prediction level. SHAP enables visual inspection of the nonlinearity and interaction between variables from a PCE model. In addition, as an alternative to Sobol indices, SHAP also quantifies the relative importance of the inputs to the output. This paper introduces a procedure to calculate SHAP values from a PCE model without explicitly building multiple PCE models. A fast and exact algorithm that enables the calculation of SHAP for high-dimensional problems is presented. The usefulness of SHAP with PCE is demonstrated on several algebraic and non-algebraic problems. • An explainable AI technique is implemented within polynomial chaos expansion (PCE). • The Shapley Additive Explanations (SHAP) enhances the explainability of PCE. • A method to analytically calculate SHAP for PCE is presented. • SHAP allows visualization of nonlinearity and interactions from the PCE model. • The method is demonstrated on three nonlinear uncertainty quantification problems. [ABSTRACT FROM AUTHOR]

Optimal non-planar wing planform geometries for a Mach 2.2 45-seat supersonic transport are generated using Kriging-based Bayesian optimisation techniques on a 22-variable, 3-objective problem for two wing-loading conditions. Sonic boom loudness, drag, and root bending moment are simultaneously minimised at a fixed lift. A Python-based framework is developed for rapid evaluation of designs in a parallel computation environment, automating conceptual geometry generation, optimised mesh griding, Euler CFD evaluation, and thermoviscous atmospheric boom propagation. An empirical turbulent skin friction coefficient is used to correct the Euler drag and an estimate of wave drag is performed. Self-organising maps and polynomial chaos expansion Shapley effects are used to explore the design variables and their associated sensitivities with respect to the design objectives. A geometric centroid-based proxy for trim is used to filter the results; feasible extreme designs and a compromise design are compared and discussed. The results indicate a trade-off between each of the three objectives due to competing design requirements, however an improvement in two of the objective functions is possible by sacrificing the remaining objective. One possible compromise solution reduces the root bending moment coefficient by 25% and the sonic boom loudness by 24% at the cost of a 5% increase in drag. [ABSTRACT FROM AUTHOR]